$\begin{array}{l}\phantom{\phantom{;}x^{4}+x^{2}+x\phantom{;}+1;}{\phantom{;}x^{3}\phantom{-;x^n}-x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{;}x^{4}+x^{2}+x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{4}+x^{2}+x\phantom{;}+1;}\underline{-x^{7}\phantom{-;x^n}-x^{5}-x^{4}-x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{7}-x^{5}-x^{4}-x^{3};}-x^{5}-x^{4}-x^{3}\phantom{-;x^n}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{4}+x^{2}+x\phantom{;}+1-;x^n;}\underline{\phantom{;}x^{5}\phantom{-;x^n}+x^{3}+x^{2}+x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}x^{5}+x^{3}+x^{2}+x\phantom{;}-;x^n;}-x^{4}\phantom{-;x^n}+x^{2}+x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{4}+x^{2}+x\phantom{;}+1-;x^n-;x^n;}\underline{\phantom{;}x^{4}\phantom{-;x^n}+x^{2}+x\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{;;\phantom{;}x^{4}+x^{2}+x\phantom{;}+1\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}2x^{2}+2x\phantom{;}\phantom{-;x^n}\\\end{array}$

$x^{3}-x-1+\frac{2x^{2}+2x}{x^4+x^2+x+1}$